# zbMATH — the first resource for mathematics

On pseudosymmetric para-Kählerian manifolds. (English) Zbl 1076.53034
A para-Kählerian manifold is a triple $$(M,J,g)$$, where $$M$$ is a differentiable manifold of dimension $$n=2m$$, $$J$$ a $$(1,1)$$-tensor field and $$g$$ a pseudo-Riemannian metric on $$M$$ satisfying the conditions $$J^2=I$$, $$g(JX,JY)=-g(X,Y)$$, $$\nabla J=0$$. Let $$R,S$$ be the curvature tensor and the Ricci tensor of $$g$$ respectively. For a $$(0,k)$$-tensor $$(k\geq 1)$$ field $$T$$ one defines the $$(0,k+2)$$-tensor field $$R\cdot T$$ by the condition $(R\cdot T)(U,V,X_1,\dots,X_k)=-\sum_{s=1}^{k} T(X_1,\dots,R(U,V)X_s,\dots,X_k)$ and the $$(0,k+2)$$-tensor field $$Q(g,T)$$ by $Q(g,T)(U,V,X_1,\dots,X_k)=-\sum_{s=1}^{k}T(X_1,\dots,(U\wedge V)X_s,\dots,X_k).$ The manifold is called semisymmetric if $$R\cdot R=0$$, Ricci-semysimmetric if $$R\cdot S=0$$, pseudosymmetric (respectively, Ricci pseudosymmetric) if there exists a function $$L: M \rightarrow \mathbb R$$ such that $$R\cdot R=LQ(g,R)$$ (respectively, $$R\cdot S=LQ(g,S)$$). Analogous definitions involving (semi or pseudo)symmetry with respect to the Bochner or the paraholomorphic projective curvature tensor can be considered.
The author proves several results where pseudosymmetry imply semi-symmetry. As an example: A pseudosymmetric para-Kählerian manifold $$(M,J,g)$$ of dimension $$>4$$ is semisymmetric. Moreover, new examples of para-Kählerian manifolds which are Ricci-semisymmetric (in dimensions $$\geq 6$$) or pseudosymmetric (in dimension 4) or Bochner-pseudosymmetric (in dimension 4) are given.

##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C55 Global differential geometry of Hermitian and Kählerian manifolds
##### Keywords:
para-Kählerian manifold; pseudosymmetric
Full Text: